Conic section

A conic section is any of the geometric figures that can arise when a plane intersects a cone. (In fact, one usually considers a "two-ended cone," that is, two congruent right circular cones placed tip to tip so that their axes align.) As is clear from their definition, the conic sections are all plane curves, and every conic section can be described in Cartesian coordinates by a polynomial equation of degree two or less.

Classification of conic sections

All conic sections fall into the following categories:

Nondegenerate conic sections

A circle is the conic section formed when the cutting plane is parallel to the base of the cone or equivalently perpendicular to the axis. (This is really just a special case of the ellipse -- see the next bullet point.)

An ellipse is formed if the cutting plane makes an angle with an axis that is larger than the angle between the slant height and the axis.

By Klaas van Aarsen - Created as a latex tikzpicturePreviously published: Not published before, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=25261046

A parabola is formed when the cutting plane makes an angle with the axis that is equal to the angle between the element of the cone and the axis.

By Klaas van Aarsen - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=25261094

An hyperbola is formed when the cutting plane makes an angle with the axis that is smaller than the angle between the element of the cone and the axis.

By Klaas van Aarsen - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=25261095

Degenerate conic sections

If the cutting plane passes through the vertex of the cone, the result is a degenerate conic section. Degenerate conics fall into three categories:

If the cutting plane makes an angle with the axis that is larger than the angle between the element of the cone and the axis then the plane intersects the cone only in the vertex, i.e. the resulting section is a single point. This is a degenerate ellipse.

If the cutting plane makes an angle with the axis equal to the angle between the element of the cone and the axis then the plane is tangent to the cone and the resulting section is a line. This is a degenerate parabola.

If the cutting plane makes an angle with the axis that is smaller than then angle between the element of the cone and the axis then the resulting section is two intersecting lines. This is a degenerate hyperbola.

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There are alternate (but equivalent) definitions of every conic section. We present them here:

Definitions of conic sections in terms of a focus and a directrix

Circle - The locus of all points equidistant from a fixed point (i.e. the directrix is infinitely far from the origin)( $e = 0$ ).
Ellipse - The locus of all points where the distance from a line (the directrix) is some number $0 < e < 1$ times the distance to some fixed point (the focus).
Parabola - The locus of all points that are an equal distance away from a point (focus) and a line (called the directrix) ( $e = 1$ ).
Hyperbola - The locus of all points where the distance from a line (the directrix) is some number $e > 1$ times the distance to some fixed point (the focus).

Definitions of conic sections in terms of foci

Circle - The locus of all points equidistant from a fixed point (i.e. the two foci are the same point).
Ellipse - The locus of all points where the sum of the distances to two points (the foci) is the same.
Parabola - The locus of all points where the sum or difference of the distances to a point (the focus) and an infinitely far point (which can be replaced by a line) is the same.
Hyperbola - The locus of all points where the absolute difference of the distances to two points (the foci) is the same.

Definitions of conic sections in terms of Cartesian coordinates

Circle - $(x-h)^{2} + (y-k)^{2} = r^2$ , where $(h,k)$ is the center of the circle, and $r$ is the radius of the circle.
Ellipse - $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ or $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ , where $(h,k)$ is the center of the ellipse, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis.
Parabola - $(x-h)^2 = 4p(y-k)$ or $(y-k)^2 = 4p(x-h)$ , where $(h,k)$ is the vertex of the parabola, and $p$ is the distance between the focus and the vertex. (Also the distance from the vertex to the directrix).
Hyperbola - $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(x-h)^2}{b^2} - \frac{(y-k)^2}{a^2} = 1$ , where $(h,k)$ is the center of the hyperbola, $a$ is the length of the semi-transverse axis, and $b$ is the length of the semi-conjugate axis.

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